Answer
We have a maximum of $3$ positive real roots and a maximum of $1$ negative real root.
Work Step by Step
We need to use Descartes' Rule.
Looking at the coefficients of the initial function, we count 3 changes in signs. Thus, we have a maximum of $3$ positive real roots.
Plug in $-x$ into the function:
$(-x)= 3(-x)^4-2(-x)^3+5(-x)^2+(-x)-9$
or, $ f(-x)= 3x^4+2x^3+5x^2-x-9$
We have one change in sign. Thus, we have a maximum of $1$ negative real root.
So, we have a maximum of $3$ positive real roots and a maximum of $1$ negative real root.
